To find the value of [tex]\(\cos 2\theta\)[/tex] given that [tex]\(\sin \theta = \frac{12}{13}\)[/tex] and that [tex]\(\theta\)[/tex] is in the first quadrant, we will use the double-angle formula for cosine and some trigonometric identities. Let's go through the solution step by step:
1. Given Information:
- [tex]\(\sin \theta = \frac{12}{13}\)[/tex]
- [tex]\(\theta\)[/tex] is in the first quadrant, implying that both [tex]\(\sin \theta\)[/tex] and [tex]\(\cos \theta\)[/tex] are positive.
2. Find [tex]\(\cos \theta\)[/tex]:
- We use the Pythagorean identity:
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1
\][/tex]
Given [tex]\(\sin \theta = \frac{12}{13}\)[/tex], we first find [tex]\(\sin^2 \theta\)[/tex]:
[tex]\[
\sin^2 \theta = \left(\frac{12}{13}\right)^2 = \frac{144}{169}
\][/tex]
- Next, solve for [tex]\(\cos^2 \theta\)[/tex]:
[tex]\[
\cos^2 \theta = 1 - \sin^2 \theta = 1 - \frac{144}{169} = \frac{169}{169} - \frac{144}{169} = \frac{25}{169}
\][/tex]
- Since [tex]\(\theta\)[/tex] is in the first quadrant, [tex]\(\cos \theta\)[/tex] is positive:
[tex]\[
\cos \theta = \sqrt{\cos^2 \theta} = \sqrt{\frac{25}{169}} = \frac{5}{13}
\][/tex]
3. Apply the Double-Angle Formula for Cosine:
- The formula for [tex]\(\cos 2\theta\)[/tex] is:
[tex]\[
\cos 2\theta = \cos^2 \theta - \sin^2 \theta
\][/tex]
- Plug in the values we found for [tex]\(\cos \theta\)[/tex] and [tex]\(\sin \theta\)[/tex]:
[tex]\[
\cos^2 \theta = \left(\frac{5}{13}\right)^2 = \frac{25}{169}
\][/tex]
[tex]\[
\sin^2 \theta = \left(\frac{12}{13}\right)^2 = \frac{144}{169}
\][/tex]
- Substitute these into the double-angle formula:
[tex]\[
\cos 2\theta = \frac{25}{169} - \frac{144}{169} = \frac{25 - 144}{169} = \frac{-119}{169}
\][/tex]
4. Conclusion:
- Therefore, the value of [tex]\(\cos 2\theta\)[/tex] is:
[tex]\[
\cos 2\theta = -\frac{119}{169} \approx -0.7041
\][/tex]
Thus, the value of [tex]\(\cos 2\theta\)[/tex] given [tex]\(\sin \theta = \frac{12}{13}\)[/tex] is [tex]\(\boxed{-0.7041}\)[/tex].
